Combining Forward and Backward Transformations

Another simple idea consists in performing a forward calculation from state i, corresponding to A, to an additional intermediate at A + AX/2, and a backward calculation from state i + 1 - which corresponds to A + AX - to the same additional intermediate. The difference in the free energies obtained from these calculations is equal to AAiyi+1. Combining (2.8) and (2.9), we obtain 

In this equation, we used the fact that the potential energy difference between the states Aj, or A + AX, and state A + A /2 is equal to AUiyi+1/2, which is a consequence of the linear form of (2.42). 

This approach is one of the oldest techniques for improving FEP calculations [36]. It is often called the simple overlap sampling (SOS) method and is usually markedly more accurate than simple averaging. It requires that one forward and one backward calculation be performed at every intermediate state. It is worth noting that no sampling is performed from the ensemble characterized by Xi+AX/2, so that the number of stages is the same as in the pure forward, or backward calculation. 

From what has been seen so far, it is obvious that the additional intermediate does not have to be located at A + AX/2, but, instead, may be chosen at any value between A and A + AX. What we would like to do is to find the location of this intermediate that minimizes the statistical error of the calculated free energy difference, AAiii+i. This problem was studied 30 years ago by Bennett [37], As it turns out, it is equivalent to calculating AAiji+1 from the formula 

are the sample sizes collected in the states % and i + T Equation (2.52) cannot be solved directly because it involves the unknown value of AAiti+1. Instead, (2.51) and (2.52) may be solved iteratively during post-simulation processing. 

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